Games, equations and dot-depth two monoids

AbstractGiven any finite alphabet A and positive integers m1,…,mk, congruences on A∗, denoted by ~(m1,…,mk) and related to a version of the Ehrenfeucht-Fraisse game, are defined. Level k of the Straubing hierarchy of aperiodic monoids can be characterized in terms of the monoids A∗/~(m1,…,mk. A natural subhierarchy of level 2 and equation systems satisfied in the corresponding varieties of monoids are defined. For |A|≥2, a necessary and sufficient condition is given for A∗/~(m1,…,mk) to be of dot-depth exactly 2. Upper and lower bounds on the dot-depth of the A∗/~(m1,…,mk) are discussed